On the Rank of the Semigroup $ T_{E} $(X)
Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ fo...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Huisheng, Pei [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2004 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer 2004 |
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| Übergeordnetes Werk: |
Enthalten in: Semigroup forum - Springer-Verlag, 1970, 70(2004), 1 vom: 26. Aug., Seite 107-117 |
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| Übergeordnetes Werk: |
volume:70 ; year:2004 ; number:1 ; day:26 ; month:08 ; pages:107-117 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00233-004-0135-z |
|---|
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OLC2044753413 |
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| 520 | |a Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5. | ||
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10.1007/s00233-004-0135-z doi (DE-627)OLC2044753413 (DE-He213)s00233-004-0135-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Huisheng, Pei verfasserin aut On the Rank of the Semigroup $ T_{E} $(X) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 2004 Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5. Topological Space Transformation Semigroup Full Transformation Full Transformation Semigroup Homeomorphism Group Enthalten in Semigroup forum Springer-Verlag, 1970 70(2004), 1 vom: 26. Aug., Seite 107-117 (DE-627)129541842 (DE-600)217500-9 (DE-576)014990733 0037-1912 nnns volume:70 year:2004 number:1 day:26 month:08 pages:107-117 https://doi.org/10.1007/s00233-004-0135-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4036 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4325 AR 70 2004 1 26 08 107-117 |
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10.1007/s00233-004-0135-z doi (DE-627)OLC2044753413 (DE-He213)s00233-004-0135-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Huisheng, Pei verfasserin aut On the Rank of the Semigroup $ T_{E} $(X) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 2004 Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5. Topological Space Transformation Semigroup Full Transformation Full Transformation Semigroup Homeomorphism Group Enthalten in Semigroup forum Springer-Verlag, 1970 70(2004), 1 vom: 26. Aug., Seite 107-117 (DE-627)129541842 (DE-600)217500-9 (DE-576)014990733 0037-1912 nnns volume:70 year:2004 number:1 day:26 month:08 pages:107-117 https://doi.org/10.1007/s00233-004-0135-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4036 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4325 AR 70 2004 1 26 08 107-117 |
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10.1007/s00233-004-0135-z doi (DE-627)OLC2044753413 (DE-He213)s00233-004-0135-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Huisheng, Pei verfasserin aut On the Rank of the Semigroup $ T_{E} $(X) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 2004 Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5. Topological Space Transformation Semigroup Full Transformation Full Transformation Semigroup Homeomorphism Group Enthalten in Semigroup forum Springer-Verlag, 1970 70(2004), 1 vom: 26. Aug., Seite 107-117 (DE-627)129541842 (DE-600)217500-9 (DE-576)014990733 0037-1912 nnns volume:70 year:2004 number:1 day:26 month:08 pages:107-117 https://doi.org/10.1007/s00233-004-0135-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4036 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4325 AR 70 2004 1 26 08 107-117 |
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10.1007/s00233-004-0135-z doi (DE-627)OLC2044753413 (DE-He213)s00233-004-0135-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Huisheng, Pei verfasserin aut On the Rank of the Semigroup $ T_{E} $(X) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 2004 Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5. Topological Space Transformation Semigroup Full Transformation Full Transformation Semigroup Homeomorphism Group Enthalten in Semigroup forum Springer-Verlag, 1970 70(2004), 1 vom: 26. Aug., Seite 107-117 (DE-627)129541842 (DE-600)217500-9 (DE-576)014990733 0037-1912 nnns volume:70 year:2004 number:1 day:26 month:08 pages:107-117 https://doi.org/10.1007/s00233-004-0135-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4036 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4325 AR 70 2004 1 26 08 107-117 |
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10.1007/s00233-004-0135-z doi (DE-627)OLC2044753413 (DE-He213)s00233-004-0135-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Huisheng, Pei verfasserin aut On the Rank of the Semigroup $ T_{E} $(X) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer 2004 Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5. Topological Space Transformation Semigroup Full Transformation Full Transformation Semigroup Homeomorphism Group Enthalten in Semigroup forum Springer-Verlag, 1970 70(2004), 1 vom: 26. Aug., Seite 107-117 (DE-627)129541842 (DE-600)217500-9 (DE-576)014990733 0037-1912 nnns volume:70 year:2004 number:1 day:26 month:08 pages:107-117 https://doi.org/10.1007/s00233-004-0135-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4036 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4325 AR 70 2004 1 26 08 107-117 |
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Huisheng, Pei |
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Huisheng, Pei |
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10.1007/s00233-004-0135-z |
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510 |
| title_sort |
on the rank of the semigroup $ t_{e} $(x) |
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On the Rank of the Semigroup $ T_{E} $(X) |
| abstract |
Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5. © Springer 2004 |
| abstractGer |
Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5. © Springer 2004 |
| abstract_unstemmed |
Abstract ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5. © Springer 2004 |
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On the Rank of the Semigroup $ T_{E} $(X) |
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